Problem: Solve for $x$ : $ 3|x - 6| - 2 = -2|x - 6| + 2 $
Solution: Add $ {2|x - 6|} $ to both sides: $ \begin{eqnarray} 3|x - 6| - 2 &=& -2|x - 6| + 2 \\ \\ { + 2|x - 6|} && { + 2|x - 6|} \\ \\ 5|x - 6| - 2 &=& 2 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 5|x - 6| - 2 &=& 2 \\ \\ { + 2} &=& { + 2} \\ \\ 5|x - 6| &=& 4 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x - 6|} {{5}} = \dfrac{4} {{5}} $ Simplify: $ |x - 6| = \dfrac{4}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 6 = -\dfrac{4}{5} $ or $ x - 6 = \dfrac{4}{5} $ Solve for the solution where $x - 6$ is negative: $ x - 6 = -\dfrac{4}{5} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& -\dfrac{4}{5} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& -\dfrac{4}{5} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{4}{5} {+ \dfrac{30}{5}} $ $ x = \dfrac{26}{5} $ Then calculate the solution where $x - 6$ is positive: $ x - 6 = \dfrac{4}{5} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& \dfrac{4}{5} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& \dfrac{4}{5} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{4}{5} {+ \dfrac{30}{5}} $ $ x = \dfrac{34}{5} $ Thus, the correct answer is $x = \dfrac{26}{5} $ or $x = \dfrac{34}{5} $.